Topological complexity is a fibrewise L–S category

Topological complexity TC ( B ) of a space B is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version TC M ( B ) of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities TC ( B ) = cat B ∗ ( d ( B ) ) + 1 and TC M ( B ) = cat B B ( d ( B ) ) + 1 , where d ( B ) = B × B is a fibrewise pointed space over B whose projection and section are given by p d ( B ) = pr 2 : B × B → B the canonical projection to the second factor and s d ( B ) = Δ B : B → B × B the diagonal. In addition, our method in studying fibrewise L–S category is able to treat a fibrewise space with singular fibres.